The Friedman-Savage piece starts with an obvious puzzle: why do people both buy lottery tickets and insurance against losses? That would seem to make them both risk-loving and risk-averse at the same time. The proffered answer is simple: part of the utility function is concave, and part is convex. Across the lower range we wish to play it safe, but above a certain margina we are willing to take gambles (by the way, here is some evidence, and why it might follow from market constraints).
For years this approach rubbed the "foundationalist Tyler" the wrong way. "Surely there is a more general approach which will allow us to derive both behaviors from a few axioms concerning risk and utility. We can’t just postulate arbitrary shifts in the curve across the utility space. Maybe both parts of the curve follow from the "temporal resolution of uncertainty," that missing variable from so much of expected utility theory. The Friedman-Savage approach will someday be seen as a diversion from the path which led to truth."
Many articles explored these routes, most notably Mark Machina’s 1982 piece on generalized expected utility theory. None of them caught on. A subsequent dose of empirical and experimental work indicated that behavior toward risk is strongly context-dependent. Neuroeconomics implied that different decisions in fact may stem from different parts of our brain, thereby challenging the assumption of a unified agent. Probably there is no overarching approach to all of the so-called violations of expected utility theory. People simply behave differently toward risk in different situations.
In other words, Friedman and Savage were ahead of their time. This is no accident, but rather it stems from Milton’s wise pragmatism, and from his general lack of interest in foundations. He also never explained "why people hold money," or "what money really is," yet he charged ahead with monetary theory and indeed monetary policy. The monetary foundationalists have been just as unsuccessful as the utility foundationalists.